Mathematical constraints on the scaling exponents in the inertial range of fluid turbulence

Abstract

The dominant view in the theory of fluid turbulence assumes that, once the effect of the Reynolds number is negligible, moments of order n of the longitudinal velocity increment, (δu), can be described by a simple power-law rζn, where the scaling exponent ζn depends on n and, except for ζ3(=1), needs to be determined. In this Letter, we show that applying Hölder's inequality to the power-law form (δu)n¯∼(rL)ζn (with r/L⪡1; L is an integral length scale) leads to the following mathematical constraint: ζ2p=pζ2. When we further apply the Cauchy–Schwarz inequality, a particular case of Hölder's inequality, to |(δu)3¯| with ζ3=1, we obtain the following constraint: ζ2≤2/3. Finally, when Hölder's inequality is also applied to the power-law form (|δu|)n¯∼(rL)ζn (this form is often used in the extended self-similarity analysis) while assuming ζ3=1, it leads to ζ2=2/3. The present results show that the scaling exponents predicted by the 1941 theory of Kolmogorov in the limit of infinitely large Reynolds number comply with Hölder's inequality. On the other hand, scaling exponents, except for ζ3, predicted by current small-scale intermittency models do not comply with Hölder's inequality, most probably because they were estimated in finite Reynolds number turbulence. The results reported in this Letter should guide the development of new theoretical and modeling approaches so that they are consistent with the constraints imposed by Hölder's inequality.